Laser Induced Control of Condensed Phase Electron Transfer

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1 Laser Induced Control of Condensed Phase Electron Transfer Rob D. Coalson, Dept. of Chemistry, Univ. of Pittsburgh Yuri Dakhnovskii, Dept. of Physics, Univ. of Wyoming Deborah G. Evans, Dept. of Chemistry, Univ. of New Mexico Vassily Lubchenko, Dept. of Chemistry, M.I.T. NH 3 NH3 CN CN H 3 N +3 + Ru N C Ru CN H 3 N NH 3 NC [polar Solvent] CN

2 q Tunneling in A -State System tropolone L> R> V(x) Proton Transfer q Multi (Double) Quantum Well Structure GaAs qalgaas GaAs e - L> R> V(x) Electron Transport In Solids q

3 N Atomic Orbitals Donor D> D> e - A> A> Acceptor Ru + Ru +3 Molecular Electron Transfer β Equivalent -state model For any of these systems: Ψ(t)> = c D (t) D> + c A (t) A> where H AA H AD c A 1 = ih d dt c A H DA H DD c D c D For a Symmetric Tunneling System: H AA = H DD = 0; H AD = H AD = (< 0)

4 Given initial preparation in D>, for a symmetric system: 1 P D (t) = 1 P A (t) = cos ( t) = ( cos( x) ) 0.5 π x t

5 Now, apply an electric field e - β E D> A> Q: How does this modify the Hamiltonian?? R 0 R A: It modifies the site energies according to -µ E Permanent dipole moment of A Thus: H = H AA H DD - E e o R/ 0 0 -e o R/

6 Analysis in the case of time-dependent E(t) = E o cosω o t Consider first the symmetric case: i d dt c A c D = µe ocosω o t Permanent dipole moment difference Letting: c A (t) = e i a sin ω ot c AI (t) ; c D (t) = e i a sin ω ot c DI (t) Where: a = µe t o µe o [ N.B.: µ (h)ω 0 E o cosω o t'dt' = ] o ω sinω o t o Thus, Interaction Picture S.E. reads: c A c D i d dt c A I c D I = 0 e i a sin ω ot 0 e -i a sin ω ot c A I c D I

7 Now note: e i b sin ωt = Σ J m (b) e i m ωt m=- So: e i a sin ω ot = Σ J m (a) e i m ω o t m=- RWA J o (a), for /ω o << 1 Thus, the shuttle frequency is renormalized to J o (a) [ ω <<1 ] NB: Trapping or localization occurs at certain E o values! a [Grossman - Hänggi, Dakhnovskii Metiu]

8 Add coupling to a condensed phase environment H ˆ = T ˆ V D (x) V A (x) - µe o cosω o t Nuclear coordinate kinetic energy Nuclear coordinate Field couples only to -level system V D (x) V A (x) ϕ D (x,t) ϕ A (x,t) x

9 Now: T+V D (x) T+V A (x) - µe o cosω o t ϕ D (x,t) ϕ A (x,t) = i d dt ϕ D (x,t) ϕ A (x,t)

10 Construction of (Diabatic) Potential Energy functions for Polar ET Systems: A D V D (x) A D V A (x)

11 A few features of classical Nonadiabatic ET Theory [Marcus, Levich-Doganadze ] Ĥ = ˆ T+V 1 (x) T+V ˆ (x) Hamiltonian non-adiabatic coupling matrix element kinetic E of nuclear coordinates (diabatic) nuclear coord. potential for electronic state Ψ(t)> = ϕ 1 (x,t) ϕ (x,t) States

12 Given initial preparation in electronic state 1 (and assuming nuclear coordinates are equilibrated on V 1 (x)) 1 ϕ 1 (x,0) x Then, P (t) = fraction of molecules in electronic state = < ϕ (x,t) ϕ (x,t)> k 1 t where k 1 = (Golden Rule) rate constant

13 In classical Marcus (Levich-Doganadze) theory, k 1 is determined by 3 molecular parameters:, E r, ε ( π ) k 1 = E r kt ½e -(E r - ε) /4E r k B T w/ E r = Reorganization Energy ; e = Reaction Heat 1 ε E r x ( π ) For the backwards Reaction: k 1 = E r kt ½e -(E r + ε) /4E r k B T

14 To obtain electronic state populations at arbitrary times, solve kinetic [ Master ] Eqns.: dp 1 (t)/dt = - k 1 P 1 (t) + k 1 P (t) dp (t)/dt = k 1 P 1 (t) - k 1 P (t) Note that long-time asymptotic [ Equilibrium ] distributions are then given by: K eq = P ( ) P 1 ( ) k 1 = e e/k B = T k 1 for Marcus formula rate constants N.B. Marcus theory for nonadiabatic ET reactions works experimentally. See: Closs & Miller, Science 40, 440 (1988)

15 Control of Rate Constants in Polar Electronic Transfer Reactions Via an Applied cw Electric Field The Hamiltonian is: V D V A ˆ H = ˆ h D 0 0 ˆ ha x + µ 1 E o cosω o t The forward rate constant is: [Y. Dakhnovskii, J. Chem. Phys. 100, 649 (1994) k D A = Re Σ J m (a) dt e ˆ ˆ i m w o t tr {ρ m=- π D -i ˆh e A t i h e t D } a = µ 1 E o / hω o 0 β

16 k D A = A D 4 ( π ) E r kt ½ Σ J m (µ 1 E o /hω o ) m=- e -(E r + ε + hmω o ) /4E r k B T Rate constants in presence of cw E-field

17 300 Schematically: Energy D A J 1 J 0 J hω o x In polar electron transfer reactions, for Reorganization Energy E r hω o (the quantum of applied laser field)

18 1 0.8 J o (a) J 1 (a) J (a) a

19 Dramatic perturbations of the one-way rate constants may be obtained by varying the laser field intensity: [Activationless reaction, E r =1eV] Forward rate constant

20 Dakhnovskii and RDC showed how this property can be used to control Equilibrium Constants with an applied cw field: D A bias e

21 Results for activationless reaction: P Γ eq = A (00) PD (00) Y. Dakhnovskii and RDC, J. Chem. Phys. 103, 908 (1995)

22 Activationless ET E r = ε = hω o = 1eV = 100 cm -1 Evans, RDC, Dakhnovskii & Kim, PRL 75, 3649 (95)

23 REALITY CHECK on coherent control of mixed valence ET reactions in polar media (1) µ E E µ Orientational averaging will reduce magnitude of desired effects [Lock ET system in place w/ thin polymer films]

24 () Dielectric breakdown of medium? To achieve resonance effects for µ = 34D E r = hω o = 1eV Electric field 10 7 V/cm Giant dipole ET complex, solvent w/ reduced E r, pulsed laser reduce likelihood of catastrophe] (3) Direct coupling of E(t) to polar solvent µ [Dipole moment of solvent molecules << Dipole moment of giant ET complex]

25 (4) hω 0 1eV ; intense fields (multiphoton) excitation to higher energy states in the ET molecule, which are not considered in the present -state model. hω 1eV hω 1eV hω 1eV hω 1eV

26 Absolute Negative Conductance in Semiconductor Superlattice - + -Keay et al., Phys. Rev. Lett. 75, 410 (1995)

27 Absolute Negative Conductance in Semiconductor Superlattice -Keay et al., Phys. Rev. Lett. 75, 410 (1995)

28 Immobilized long-range Intramolecular Electron Transfer Complex: D Tethered Alkane Chain molecule A (hω) k E (Inert) Substrate

29 Light Absorption by Mixed Valence ET Complexes in Polar Solvents: Incident Photons hω ο hω ο hω ο Ru + Ru +3 Solvent hω hω ο ο hω ο Absorption cross section Kabs ( ) = ω ο # Absorbed Transmitted Photons Photons # Incident Photons

30 Absorption Cross Section Formula: (1) ), ( 1 dt a d E E k abs ο ο ω = () ), ( ) ( 1 ) ( 1 t U n t U n dt a E d eq eq + = ω ο ( ) ( ) + ± ± = = T k E m E T k E m E a J m T k E t U B r r B r r m m B r 4 exp 4 exp (3) ) ( , ο ο ο ω ε ω ε ω π ħ ħ ħ ħ ο ο ο ω µ ħ E a =

31 Hush Absorption Spectrum σ ( ω L ) = # Absorbed Photons # Incident Photons ω L 1 [ FCF] Marcus Gaussian w/ ħ ω 1 = effective "transition dipole moment" = µ? << ħω res µ permanent dipole moment difference

32 Barrierless PRL 77, 917 (1996) J. Chem. Phys. 105, 9441 (1996) Hush Absorption = µ οe ħω ο ο

33 absorption emission Stimulated Emission using two incoherent lasers J. Chem. Phys. 109, 691 (1998)

34 J. Chem. Phys. 109, 691 (1998)

12.2 MARCUS THEORY 1 (12.22)

12.2 MARCUS THEORY 1 (12.22) Andrei Tokmakoff, MIT Department of Chemistry, 3/5/8 1-6 1. MARCUS THEORY 1 The displaced harmonic oscillator (DHO) formalism and the Energy Gap Hamiltonian have been used extensively in describing charge